POJ 3641:Pseudoprime numbers

费马小定理与伪素数判断

最新推荐文章于 2022-03-03 15:49:58 发布

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最新推荐文章于 2022-03-03 15:49:58 发布

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原文链接：http://www.cnblogs.com/Friends-A/p/9309070.html

本文介绍了一种基于费马小定理的算法，用于判断一个合数是否为特定类型的伪素数（即对于某个基a，a^p mod p = a但p并非素数）。通过快速幂运算实现高效计算，并给出完整的C++代码实现。

Description

Fermat’s theorem states that for any prime number p and for any integer a > 1, ap = a (mod p). That is, if we raise a to the pth power and divide by p, the remainder is a. Some (but not very many) non-prime values of p, known as base-a pseudoprimes, have this property for some a. (And some, known as Carmichael Numbers, are base-a pseudoprimes for all a.)
Given 2 < p ≤ 1000000000 and 1 < a < p, determine whether or not p is a base-a pseudoprime.

Input

Input contains several test cases followed by a line containing “0 0”. Each test case consists of a line containing p and a.

Output

For each test case, output “yes” if p is a base-a pseudoprime; otherwise output “no”.

Sample Input
3 2
10 3
341 2
341 3
1105 2
1105 3
0 0

Sample Output
no
no
yes
no
yes
yes

题意：不是素数的数p，且a^p对p取模等于a，输出yes，其他的输出no。
直接暴力，不要打表，打表数组开不到1e9，暴力不会超时

#include<stdio.h>
#include<math.h>
#define ll long long
ll Pow(ll a,ll b)
{
    ll res=1;
    ll c=b;
    while(b>0)
    {
        if(b&1) res=res*a%c;
        a=a*a%c;
        b>>=1;
    }
    return res;
}
ll su(ll n)
{
    for(ll i=2;i*i<=n;i++)  
    {
        if(n%i==0)
        return 0;
    }
    return 1;  
}
int main()
{
    ll p,a,flag=0;
    while(~scanf("%lld%lld",&p,&a))
    {
        if(a==0&&p==0) break;
        if(su(p)==0) flag+=1;
        if(flag!=0&&Pow(a,p)==a) printf("yes\n");
        else printf("no\n");
        flag=0;
    }
    return 0;
}